Operator moment dilations as block operators

Let $\mathcal{H}$ be a complex Hilbert space and let $\big\{A_{n}\big\}_{n\geq 1}$ be a sequence of bounded linear operators on $\mathcal{H}$. Then a bounded operator $B$ on a Hilbert space $\mathcal{K} \supseteq \mathcal{H}$ is said to be a dilation of this sequence if \begin{equation*} A_{n} = P_{\mathcal{H}}B^{n}|_{\mathcal{H}} \; \text{for all}\; n\geq 1, \end{equation*} where $P_{\mathcal{H}}$ is the projection of $\mathcal{K}$ onto $\mathcal{H}.$ The question of existence of dilation is a generalization of the classical moment problem. We recall necessary and sufficient conditions for the existence of self-adjoint, isometric and unitary dilations and present block operator representations for these dilations. For instance, for self-adjoint dilations one gets block tridiagonal representations similar to the classical moment problem. Given a positive invertible operator $A$, an operator $T$ is said to be in the $\mathcal{C}_{A}$-class if the sequence $\{A^{-\frac{1}{2}}T^nA^{-\frac{1}{2}}:n\geq 1\}$ admits a unitary dilation. We identify a tractable collection of $\mathcal{C}_A$-class operators for which isometric and unitary dilations can be written down explicitly in block operator form. This includes the well-known $\rho$-dilations for positive scalars. Here the special cases $\rho =1$ and $\rho =2$ correspond to Sch\"{a}ffer representation for contractions and Ando representation for operators with numerical radius not more than one respectively.